Regarding the considerations of using a differential scope, and whether you need to use a laptop that is connected to the mains or not, we need to be clear on what the PicoScope 3425 can do and what it can't do.

It CAN isolate the input section of the channels from the rest of the oscilloscope, and therefore from the ground reference that the PicoScope uses, so that you can make measurements that are with reference to a voltage that is not ground. This can include measurements of up to 400V, when combining the dfferential voltage measured with the common mode range being used, as long as the electrical energy that can be discharged between the circuit being measured and and the measurement equipment, would not be enough to damage the equipment. You can read more about what CAN do here: https://www.picotech.com/oscilloscope/3 ... 5-overview

What it CANNOT do is protect the internals of the PicoScope and the equipment connect via USB to the PicoScope from damaging electrical currents and high energy impulse. IF YOU ARE INTENDING TO USE THIS PICOSCOPE WITH HIGH VOLTAGE EQUIPMENT, THEN THAT EQUIPMENT MUST HAVE LIMITED CURRENT, AND LIMITED ENERGY IMPULSE CAPABILITY. So, for instance, you could measure the circuitry of Copier machines, that have high voltages, but low currents, and are not directly connected to electrical mains supplies, but are connected through isolating transformers.

To answer your question about the ground socket on the PicoScope, and following on from your previous question:

You can use the differential probes that come with the scope to measure the potential difference between 2 points in a circuit, or system even if neither of them are at ground potential. The probe lead connects to a channel input and the flying banana plug connects to the banana socket for that channel to provide a ground connection for the screened probe lead (see image 1).

The banana socket also provides a connection between scope ground and probe ground for a single-ended probe when using the single ended to differential adapter that comes with the scope (see image 2). The adapter screws onto the single-ended scope lead and the input channel of the scope, while the flying banana plug on the adapter just plugs into the banana socket. This allows single ended measurements to be displayed on the differential scope.

It's not necessary to use a laptop that is battery powered and therefore floating in terms of voltage levels, because you shouldn't be taking measurements from a source that could potentially include high energy impulses that might damage the scope and laptop (and that wouldn't be a good way to protect them anyway).